
Consider an axion background \(\bar{b}(x)\)  in the FRW background which is linear in (cosmic) time and satisfies the conformal invariance conditions in non-critical strings to be a background (see [1]}). This background decouples from \(\tilde{b}\)  since the \(\bar{b} \tilde{b}\)  term vanishes (provided \(\int d^{4}x\ \partial ^{0} \bar{b}\)  vanishes).The term \(\int d^{4}x\sqrt{-g} \partial _{\mu } \bar{b} J^{5\mu }\)  gives the model of leptogenesis represented by (REF ) with a constant axion background. We should sound a word of caution since the constant axion background may not be solution of superstring theory (in the presence of fermions). Even if it were a solution, the constant would need to be fine tuned.
This case has another use , besides leptogenesis, which is for axionic dark matter and a solution of the strong CP problem [2]}.
In terms of torsionless quantities the axial current anomaly is
\(d{\bf {^{\ast }J^{5}}}=-\frac{\alpha _{QED} Q^{2}}{\pi } {\bf {F}}\wedge {\bf {F}}-\frac{\alpha _{s} N_{q}}{2\pi } Tr\left( {\bf {G}}\wedge {\bf {G}}\right) -\frac{N_{f}}{8\pi ^{2} } {\bf {R}}^{ab}\wedge {\bf {R}}_{ab}\) 
where \(\alpha _{QED}\)  and \(\alpha _{s}\)  are the fine structure constants for electromagnetism and quantum chromodynamics respectively, \(N_f\)  is the number of fermion flavours and \(Q^{2}=\sum _{f} Q^{2}_{f}\)  where \(Q_f\)  is the electric charge of the fermion with flavour \(f\) , \(F\) , \(G\)  and \(R\)  are the field strength forms of QED, QCD and gravity. The resulting effective action is:
\(S_{eff}&=&S_{0}-\frac{1}{2f^{2}_{b}} \int J^{5}\wedge \ast J^{5}-\frac{\alpha _{QED} }{\pi f_{b}} \int bF\wedge F\\& &-\int \frac{1}{2} db\wedge \ast db-\frac{1}{8\pi ^{2} } \int \left( \Theta +\frac{N_{f}}{f_{b}} b\right) R^{ac}\wedge R_{ac}\nonumber \\&& -\frac{\alpha _{s} }{2\pi } \int \left( \theta +\frac{N_{q}}{f_{b}} b\right) Tr\left[ G\wedge G\right] \nonumber \) 
The constants \(\Theta \)  and \(\theta \)  are associated with topological terms and have been kept in for completeness (and classically do not affect the equations of motion). \(S_{0}\)  represents the Einstein gravitational interactions, non-gravitational gauge interactions and Yukawa interactions of matter. This model embodies the links that we mentioned in the introduction, and the \(b\)  field is at the heart of the connections.
Using the action in (31) we have explored leptogenesis. Recently the formulation used in (REF ) has also been investigated in connection with axionic dark matter. Although the model has features similar to that of the QCD axion, the gravitational axion couples to the Pontryagin densities of all the gauge interactions in the model. The model could, for example, be a generalisation of the Standard Model above the electroweak transition; then all the gauge fields would have couplings to the axion through their Pontryagin density. Another feature is the repulsive current -current interaction. This would be a contribution to the de Sitter nature of the Universe which is however suppressed by the factor of \(f_b^2\)  where \(f_b=\frac{M_p}{\sqrt{3\pi }}\approx 4\times 10^{18}GeV\)  (and \(M_p\)  is the Planck mass). It has been estimated [3]} that this gravitational axion would have a mass of order \(10^{-12 }eV\) . Axions can have a range of masses and such a mass has not been ruled out.

In the scenario for the Kalb-Ramond torsion model that we have just discussed, we consider backgrounds of \(B_{0}\)  (\(=\partial _{0}b\) ) which are constant but then typically we argue that the \(B_{0}\)  has diminished from the time of freeze-out of the sterile neutrino in our model to values which are compatible with current bounds for CPTV. It has more recently been considered that \(B_{0}\)  can be time or temperature \(T\)  dependent.In the radiation era the scale factor \(a(t)\)  in the Friedmann-Robertson-Walker metric goes with time like \(a(t)\sim t^{1/2}\) .
From (31) we find the equation of motion of \(b\)  to be
\(\partial _{\mu } \left[ \sqrt{-g} \left( \frac{8}{3\kappa ^{2} } \partial _{\mu } b-J^{5\mu }\right) \right] =0.\) 
For the flat FRW metric \(\sqrt{-g} \sim a^{3}\sim T^{-3}\)  where \(g\)  is the determinant of this metric and \(a(t)\)  is the scale factor. We consider for (REF ) only dependence on time.
On assuming that a chiral current condensate has not formed we find that
\(B_{0}\left( T\right) \sim T^{3}.\) 
If we assume that the decoupling temperature \(T_{D}=O\left( 100\right) \)  TeV, as was the case for constant \(B_{0}\)  analysis, we are in the radiation era. In the general case
\(\frac{8}{3\kappa ^{2} } B_{o}\left( T\right) -\langle J^{5}_{0}\rangle _{T} =A^{\prime }T^{3}\) 
where \(A^{\prime }\)  is a constant of integration. We can evaluate the thermal average \(\langle J^{5}_{0}\rangle _{T}\)  using equilibrium thermodynamics in the presence of a background field \(B_{0}(T)\) . Let us write
\(J^{5}_{0}=\left< J^{5}_{0}\right>_{T} +\sum _{i=fermions} \psi ^{\dag }_{i} \gamma ^{5} \psi _{i}.\) 
The fermion part of the Lagrangian is
\(L^{F}=\sqrt{-g} \bar{\psi } \left[ i\gamma ^{\mu } \partial _{\mu } -m1-\left( B_{0}\left( T\right) -\mu _{5} \left( T\right) \right) \gamma ^{0} \gamma ^{{}^{5}} +\mu \left( T\right) \gamma ^{0} \right] \psi +\ldots \) 
with \(\mu _{5} \left( T\right) =\frac{3\kappa ^{2} }{8} \left< J^{5}_{0}\right>_{T} .\) The \(\cdots \)  denotes current-current interactions which are highly suppressed and will be ignored. A chemical potential \(\mu \)  has been added for the quarks. For leptons, since lepton number is not conserved, the chemical potential is 0. We will consider the situation when there are no external electromagnetic fields. The dispersion relation for a fermion of mass \(m\)  and helicity \(\lambda _{r}\)  is
\(E_{r}=\sqrt{m^{2}+\left( \left| \vec{p} \right| +\lambda _{{}_{r}} B_{0}\right)^{2} } .\) 
The thermal distributions \(f(E)\)  depend on the chemical potential:
\(f\left( E\right) =\frac{1}{\exp \left( \frac{E-\mu }{T} \right) +1} .\) 
For a particle with (left/right) handedness \(\mu =\mu _{L\left( R\right) } =\mu -\left( +\right) \mu _{5} \) .The thermal distributions are used in the calculation of \(\left< J^{5}_{0}\right>_{T}\) . For high temperatures \(T\ge T_{D}\) , where \(T_D\)  is the decoupling temperature of the sterile neutrino, we can consider massless fermions and so \(E_{r}=\left| \left| \vec{p} \right| +\lambda _{r} B_{0}\right| \) . For antiparticles \(E\rightarrow -E\)  and \(\lbrace \mu , \mu _{5} \rightarrow -\mu , -\mu _{5} \rbrace \) . The thermal average \(\left< J^{5}_{0}\right>_{T}\)  has contributions from all species of fermions, both leptons and quarks.
\(\left< J^{5}_{0}\right>_{T} & = & \sum _{i} \left< \bar{\psi }_{i} \gamma ^{0} \gamma ^{{} 5} \psi _{i} \right>_{T} = {}^{}_{}\sum _{i,\lambda } \left[ \left( n^{eq}_{R}-\bar{n}^{eq}_{R} \right) -\left( n^{eq}_{L}-\bar{n}^{eq}_{L} \right)\right]_{i,\lambda } \\& = & \sum _{i}\lbrace  \left[ \left( n^{eq}_{R}-\bar{n}^{eq}_{R} \right) -\left( n^{eq}_{L}-\bar{n}^{eq}_{L} \right)\right]_{\lambda =-1 }+ \left[ \left( n^{eq}_{R}-\bar{n}^{eq}_{R} \right) -\left( n^{eq}_{L}-\bar{n}^{eq}_{L} \right)\right]_{\lambda =1 } \rbrace \nonumber \\& = & \sum _{i}\lbrace  \left[ \left( n^{eq}_{R}-\bar{n}^{eq}_{R} \right) -\left( n^{eq}_{L}-\bar{n}^{eq}_{L} \right)\right]_{\lambda =-1 }+ \left[ \left( n^{eq}_{L}-\bar{n}^{eq}_{L} \right) -\left( n^{eq}_{R}-\bar{n}^{eq}_{R} \right)\right]_{\lambda =-1 }\rbrace  \nonumber \\& = & 0 \nonumber \) 
where \(n^{eq} (\bar{n}^{eq})\)  is the thermal equilibrium number density of particles and antiparticles and the sum is over all species of fermions as well as the helicities. Hence the thermal axial current expectation vanishes and

\( B_{0}\left( T\right) \simeq AT^{3},\ \ \ \ A=\frac{3\kappa ^{2} }{8} A^{\prime }\) 
